I am trying to find a short story I read in an early 1980s (or maybe late 1970s) anthology. I probably read all the stories in the book, but only one made a real impression; I think it was the first story in the book. If I recall correctly, it had a cover with a bunch of crudely drawn cartoon figures, similar in style to the cartoons James Thurber used to illustrate his stories and books. (Is James Thurber's cartoon book about the orchestra on topic here as fantasy?)

The key element of the story was this: The main character (who I think narrated the story) somehow knew (and was friends with) everyone on Earth. He mentions knowing a girl in Israel, and everyone in Salt Lake City, which he has never visited. The people around him blow this weirdness off... until he finds that he "knows" some new people, who were not simply born like the many babies he has come to know telepathically. These new people just appeared somehow. Then the people around him ask him where the alien arrivals are. They reveal that he is actually a telepathic android. His brain is not in his head, but is a room-sized construction in the basement. He was built to telepathically identify where the alien invasion was taking place. However, since he considers himself to be "friends" with everyone he knows, including the aliens, he refuses to divulge their location. The scientists threaten to disassemble his brain, but he holds firm, even as they start to take his brain apart.

in an early 1980s (or maybe late 1970s) anthology. I probably read all the stories in the book, but only one made a real impression; I think it was the first story in the book. If I recall correctly, it had a cover with a bunch of crudely drawn cartoon figures, similar in style to the cartoons James Thurber used to illustrate his stories and books.

The main character (who I think narrated the story) somehow knew (and was friends with) everyone on Earth. He mentions knowing a girl in Israel, and everyone in Salt Lake City, which he has never visited.

Anthony Trotz is the main character; he knows everyone in the world, but not as friends ("I haven't a real friend in the lot"); the story is told from his point of view, but he is not the narrator:

Anthony had a reason for questioning the four men since (as a new thing that had come to him) he knew everybody. He knew everyone in Salt Lake City, where he had never been. He knew everybody in Jebel Shah, where the town is a little amphitheater around the harbor; and in Batangas and Weihai. He knew the loungers around the end of the Galata bridge in Istanbul, and the porters in Kuala Lumpur. He knew the tobacco traders in Plovdiv, and the cork cutters of Portugal. He knew the dock workers in Djibouti, and the glove makers in Prague. He knew the vegetable farmers around El Centro, and the muskrat trappers of Barataria Bay. He knew the three billion people of the world by name and face, and with a fair degree of accuracy.

As for the "girl in Israel", maybe you're thinking of the one in Beirut:

He spoke to Wellington who also worked in his room. "I know a girl in Beirut who is just going to bed. It is evening there now, you know."

The people around him blow this weirdness off...

It was no good trying to tell things to Wellington. Wellington never listened. And then Anthony got a summons to Colonel Peter Cooper, which always increased his apprehension.

"Anthony," said the Colonel, "I want you to tell me if you discern anything unusual. That is really your job, to report anything unusual. The other, the paper shuffling, is just something to keep your hands busy. Now tell me clearly if anything unusual has come to your notice."

"Sir, it has." And then he blurted it out. "I know everybody. I know everybody in the world. I know them all in their billions, every person. It has me worried sick."

"Yes, yes, Anthony. But tell me, have you noticed anything odd? It is your duty to tell me if you have."

"But I have just told you! In some manner I know every person in the world. I know the people in Transvaal, I know the people in Guatemala. I know everybody."

"Yes, Anthony, we realize that. And it may take a little getting used to. But that isn't what I mean. Have you, besides that thing that seems out of the way to you, noticed anything unusual, anything that seems out of place, a little bit wrong?"

until he finds that he "knows" some new people, who were not simply born like the many babies he has come to know telepathically. These new people just appeared somehow.

And then it was that something just a little bit unusual did happen, something not quite right, a small thing. But the Colonel had told him to report anything about anything, no matter how insignificant, that struck him as a little queer.

It was just that with all the people in his head, and the arrivals and departures, there was a small group that was not of the pattern. Every minute hundreds left by death and arrived by birth. And now there was a small group, seven persons; they arrived into the world, and they were not born into the world.

So Anthony went to tell Colonel Cooper that something had occurred to his mind that was a little bit odd.

Then the people around him ask him where the alien arrivals are.

"You surely took your time, Anthony. Tell me at once what it is and where. The reaction was registered, but it would take us hours to pinpoint its source without your help. Now then, explain as calmly as you can what you felt or experienced. Or, more to the point, where are they?"

They reveal that he is actually a telepathic android. His brain is not in his head, but is a room-sized construction in the basement.

They went into the barred area, down into the bowels of the main building of the center. And they looked at the brain of Anthony Trotz, a restricted person in its special meaning.

"It is the largest in the world," said Colonel Cooper.

"How large?"

"A little over twelve hundred cubic meters."

"What a brain! And it is mine?"

"You share it with others. But, yes, it is yours. You have access to its data. You are an adjunct to it, a runner for it, an appendage, inasmuch as you are anything at all."

He was built to telepathically identify where the alien invasion was taking place.

"The purpose," said Colonel Cooper, "was to notice anything just a little peculiar in the auras and the persons they represent, anything at all odd in their comings and goings. Anything like what you have come here to report to me."

"Like the seven persons who recently arrived in the world, and not by way of birth?"

"Yes. We have been expecting the first of the aliens for months. We must know their area, and at once. Now tell me."

However, since he considers himself to be "friends" with everyone he knows, including the aliens, he refuses to divulge their location.

It's rather that the earth people are not his friends:

The area is quite near. If the Colonel were not burdened with a mind, he would be able to think more clearly. He would know that cruel children and dogs love to worry what is not human, and that all the restricted persons for this area are accounted for. He would know that they are worrying one of the aliens in the street below, and that is the area that is right for my consciousness.

I wonder if they will be better masters?

The scientists threaten to disassemble his brain, but he holds firm, even as they start to take his brain apart.

And aloud he said to the Colonel:

"I will not tell you."

"Then we'll have you apart and get it out of you mighty quick."

"How quick?"

"Ten minutes."

"Time enough," said Tony.

For he knew them now, coming in like snow. They were arriving in the world by the hundreds, and not arriving by birth.

That's certainly the story. I remember that we had a copy of Strange Gifts, and that cover seems vaguely familiar. I wonder whether I confused it in my mind with another SF book with more cartoonish cover art.
– BuzzJul 18 '17 at 13:55